Problem 66 - Calculus and its applications

Question

Explain the meaning of the first partial derivatives of a function of two variables in terms of slopes of tangent lines.

Step-by-Step Solution

Answer

Partial derivatives measure the slope of the tangent line to the function's curve in the direction of each variable.

Step 1: Understanding Partial Derivatives

Partial derivatives measure the rate of change of a function with respect to one of its variables, keeping all other variables constant. For a function of two variables, \( f(x, y) \), there are usually two partial derivatives: \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \).

Step 2: Partial Derivative with Respect to x

The partial derivative \( \frac{\partial f}{\partial x} \) represents the rate of change of the function \( f \) in the direction of the x-axis. Geometrically, this is the slope of the tangent line to the curve formed by the function when the y-variable is held constant. It tells us how much \( f \) changes as only \( x \) changes.

Step 3: Partial Derivative with Respect to y

Similarly, the partial derivative \( \frac{\partial f}{\partial y} \) represents the rate of change of the function \( f \) in the direction of the y-axis. This is the slope of the tangent line to the curve formed by the function when the x-variable is held constant. It indicates how \( f \) changes as only \( y \) changes.

Key Concepts

Function of Two Variables

In mathematics, when we talk about a function of two variables, we often refer to a surface in a three-dimensional space. Think of each point on this surface as having an input pair \(x, y\) and an output, \(f(x, y)\), which represents the height of the surface at that point. Each input variable influences the function’s output differently, giving rise to rich interactions between the variables.

For instance, consider a function \(f(x, y) = x^2 + y^2\). This function combines the effects of both \(x\) and \(y\). Here, the term \(x^2\) affects the output through \(x\), while \(y^2\) influences it through \(y\).

Functions of two variables are crucial in fields like physics and economics, where multiple factors often interact. By studying these, one gains insight into how changes in inputs affect a given scenario or phenomenon.

Tangent Line Slopes

The slope of a tangent line helps in understanding how a function behaves locally. For a function of two variables, tangent line slopes are revealed through partial derivatives.

Partial derivatives symbolize the slopes of these tangent lines in the direction of each input variable.

• \( \frac{\partial f}{\partial x} \): This partial derivative shows the slope of the tangent line along the plane parallel to the x-axis, keeping \(y\) constant. It tells us how much the function value \(f(x, y)\) changes for small changes in \(x\).
• \( \frac{\partial f}{\partial y} \): This partial derivative gives us the slope of the tangent line when moving parallel to the y-axis, holding \(x\) steady. This indicates the change in function value as \(y\) varies slightly.

The steeper the tangent line in any given direction, the more sensitive the function is to changes in that variable. Understanding these slopes allows for the calculation of properties like local maximums and minimums.

Rate of Change

The rate of change is a fundamental aspect of calculus that describes how a function alters in response to changes in its variables. For functions of two variables, the rate of change in each direction is quantified through partial derivatives.

When we say rate of change, we essentially mean the speed at which the function's output responds to alterations in one input variable, assuming others are constant.

• The partial derivative \( \frac{\partial f}{\partial x} \) captures this rate by showing how responsive \(f(x, y)\) is to changes in \(x\), measuring the 'motion' along the x-axis.
• Conversely, \( \frac{\partial f}{\partial y} \) records the functioning of \(y\) when \(x\) is unchanging, thus detailing the 'rate' along the y-axis.

This concept is vital for real-world applications. Understanding these changes helps in predicting and adjusting for phenomena such as atmospheric pressure variations and economic growth indicators. Rate of change, therefore, provides a lens through which dynamic systems may be analyzed and interpreted.